Higgs Mode in the $d$-Wave Superconductor ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+x}$ Driven by an Intense Terahertz Pulse

We investigate the terahertz (THz)-pulse-driven nonlinear response in the $d$-wave cuprate superconductor ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+x}$ (Bi2212) using a THz pump near-infrared probe scheme in the time domain. We observe an oscillatory behavior of the optical reflectivity that follows the THz electric field squared and is markedly enhanced below $T_c$. The corresponding third-order nonlinear effect exhibits both $A_{1g}$ and $B_{1g}$ symmetry components, which are decomposed from polarization-resolved measurements. A comparison with a BCS calculation of the nonlinear susceptibility indicates that the $A_{1g}$ component is associated with the Higgs mode of the $d$-wave order parameter.

Figure 1

(a) A geometry for the THz pump-optical probe measurements. (b) A schematic ${\mathrm{CuO}}_2$ plane, on which the pump (${\theta }_{\mathrm{pump}}$) and probe (${\theta }_{\text{probe}}$) polarization angles are defined relative to the $\mathrm{Cu}─\mathrm{O}$ bond. (c) THz pulse-induced transient reflectivity change $\mathrm{\Delta }R/R$ at ${\theta }_{\text{probe}}=0°$ as a function of the delay time at typical temperatures for OP90. The top panel shows the waveform of the squared THz $E$ field. (d) Probe polarization dependence (circles) of the amplitude of $\mathrm{\Delta }R/R$ at fixed delays at various temperatures for OP90 when ${\theta }_{\mathrm{pump}}=0°$. Curves show the fitting with a form $A+B\cos (2{\theta }_{\text{probe}})$.

Figure 2

For OP90, temperature dependences of the $A_{1g}$ (a) and $B_{1g}$ (b) components of $\mathrm{\Delta }R/R$ are displayed with red dashed lines indicating $T_c$. (c) The $A_{1g}$ and $B_{1g}$ components against the delay time at 10 K with fitting curves. (d) Temperature dependences of the $A_{1g}$ decaying component (blue curve), the $A_{1g}$ THz Kerr component (red curve), and the $B_{1g}$ THz Kerr component (green curve).

Figure 3

(a)–(c) Temporal behavior of the $A_{1g}$ and $B_{1g}$ components as compared for UD74, OP90, and OD78 at 10 K. Red and blue curves represent the $A_{1g}$ THz Kerr component and decaying component, respectively, while black lines are total fitting curves. (d) Doping dependence of the ratio of the amplitude of the $B_{1g}$ and the $A_{1g}$ THz Kerr components at 10 K (red circles; left axis) with $T_c$ (blue; right axis) for all the samples studied. The hole concentration $p$ is determined from $T_c$ with Presland and Tallon’s equation [43].

Figure 4

Numerical result for the CDF contribution to ${\chi }^{(3)}(\omega ;\omega ,+\mathrm{\Omega },-\mathrm{\Omega })$ for various symmetry components.